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\begin{document}
\title{Topologies on Algebraic Spaces}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
In this chapter we introduce some topologies on the
category of algebraic spaces. Compare with the material in \cite{SGA1},
\cite{Ner}, \cite{LM-B} and \cite{Kn}.
Before doing so we would like to point out that there
are many different choices of sites (as defined in
Sites, Definition \ref{sites-definition-site}) which give rise to
the same notion of sheaf on the underlying category. Hence
our choices may be slightly different from those in the references
but ultimately lead to the same cohomology groups, etc.
\section{The general procedure}
\label{section-procedure}
\noindent
In this section we explain a general procedure for producing the
sites we will be working with. This discussion will make little or
no sense unless the reader has read
Topologies, Section \ref{topologies-section-procedure}.
\medskip\noindent
Let $S$ be a base scheme.
Take any category $\Sch_\alpha$ constructed as in
Sets, Lemma \ref{sets-lemma-construct-category} starting with
$S$ and any set of schemes over $S$ you want to be included.
Choose any set of
coverings $\text{Cov}_{fppf}$ on $\Sch_\alpha$ as in
Sets, Lemma \ref{sets-lemma-coverings-site}
starting with the category $\Sch_\alpha$ and the class of fppf
coverings. Let $\Sch_{fppf}$ denote the big fppf site so
obtained, and let $(\Sch/S)_{fppf}$ denote the corresponding
big fppf site of $S$. (The above is entirely as prescribed in Topologies,
Section \ref{topologies-section-fppf}.)
\medskip\noindent
Given choices as above the category of algebraic spaces over $S$
has a set of isomorphism classes. One way to see this is to use the
fact that any algebraic space over $S$ is of the form $U/R$ for
some \'etale equivalence relation $j : R \to U \times_S U$ with
$U, R \in \Ob((\Sch/S)_{fppf})$, see
Spaces, Lemma \ref{spaces-lemma-space-presentation}.
Hence we can find a full subcategory $\textit{Spaces}/S$ of the category of
algebraic spaces over $S$ which has a set of objects
such that each algebraic space is isomorphic to an object of
$\textit{Spaces}/S$. We fix a choice of such a category.
\medskip\noindent
In the sections below, given a topology $\tau$, the big site
$(\textit{Spaces}/S)_\tau$ (resp.\ the big site $(\textit{Spaces}/X)_\tau$
of an algebraic space $X$ over $S$)
has as underlying category the category $\textit{Spaces}/S$
(resp.\ the subcategory $\textit{Spaces}/X$ of $\textit{Spaces}/S$, see
Categories, Example \ref{categories-example-category-over-X}).
The procedure for turning this into a site is as usual by defining a
class of $\tau$-coverings and using
Sets, Lemma \ref{sets-lemma-coverings-site}
to choose a sufficiently large set of coverings which defines the topology.
\medskip\noindent
We point out that the {\it small \'etale site $X_\etale$
of an algebraic space $X$} has already been defined in
Properties of Spaces, Definition
\ref{spaces-properties-definition-etale-site}.
Its objects are schemes \'etale over $X$, of which there are plenty
by definition of an algebraic spaces. However,
a more natural site, from the perspective of this chapter (compare
Topologies, Definition \ref{topologies-definition-big-small-etale})
is the site $X_{spaces, \etale}$ of
Properties of Spaces, Definition
\ref{spaces-properties-definition-spaces-etale-site}.
These two sites define the same topos, see
Properties of Spaces, Lemma \ref{spaces-properties-lemma-compare-etale-sites}.
We will not redefine these in this chapter; instead we will simply
use them.
\medskip\noindent
Finally, we intend not to define the Zariski sites, since these do not
seem particularly useful (although the Zariski topology
is occasionally useful).
\section{Fpqc topology}
\label{section-fpqc}
\noindent
We briefly discuss the notion of an fpqc covering of algebraic spaces.
Please compare with
Topologies, Section \ref{topologies-section-fpqc}.
We will show in
Descent on Spaces,
Proposition \ref{spaces-descent-proposition-fpqc-descent-quasi-coherent}
that quasi-coherent sheaves descent along these.
\begin{definition}
\label{definition-fpqc-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
An {\it fpqc covering of $X$} is a family of morphisms
$\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces
such that each $f_i$ is flat and such that for every affine scheme
$Z$ and morphism $h : Z \to X$ there exists a standard fpqc covering
$\{g_j : Z_j \to Z\}_{j = 1, \ldots, n}$ which refines the family
$\{X_i \times_X Z \to Z\}_{i \in I}$.
\end{definition}
\noindent
In other words, there exists indices $i_1, \ldots, i_n \in I$ and
morphisms $h_j : U_j \to X_{i_j}$ such that
$f_{i_j} \circ h_j = h \circ g_j$. Note that if $X$ and all $X_i$ are
representable, this is the same as a fpqc covering of schemes by
Topologies, Lemma \ref{topologies-lemma-fpqc-covering-affines-mapping-in}.
\begin{lemma}
\label{lemma-fpqc}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is an fpqc covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is an fpqc covering and for each
$i$ we have an fpqc covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is an fpqc covering.
\item If $\{X_i \to X\}_{i\in I}$ is an fpqc covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is an fpqc covering.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) is clear. Consider $g : X' \to X$ and
$\{X_i \to X\}_{i\in I}$ an fpqc covering as in (3). By
Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-base-change-flat}
the morphisms $X' \times_X X_i \to X'$
are flat. If $h' : Z \to X'$ is a morphism from an affine scheme
towards $X'$, then set $h = g \circ h' : Z \to X$. The assumption
on $\{X_i \to X\}_{i\in I}$ means there exists a standard fpqc covering
$\{Z_j \to Z\}_{j = 1, \ldots, n}$ and morphisms $Z_j \to X_{i(j)}$ covering
$h$ for certain $i(j) \in I$. By the universal property of the fibre product
we obtain morphisms $Z_j \to X' \times_X X_{i(j)}$ over $h'$ also.
Hence $\{X' \times_X X_i \to X'\}_{i\in I}$ is an fpqc covering.
This proves (3).
\medskip\noindent
Let $\{X_i \to X\}_{i\in I}$ and $\{X_{ij} \to X_i\}_{j\in J_i}$ be as
in (2). Let $h : Z \to X$ be a morphism from an affine scheme towards $X$.
By assumption there exists a standard fpqc covering
$\{Z_j \to Z\}_{j = 1, \ldots, n}$ and morphisms $h_j : Z_j \to X_{i(j)}$
covering $h$ for some indices $i(j) \in I$. By assumption there exist
standard fpqc coverings
$\{Z_{j, l} \to Z_j\}_{l = 1, \ldots, n(j)}$
and morphisms $Z_{j, l} \to X_{i(j)j(l)}$ covering
$h_j$ for some indices $j(l) \in J_{i(j)}$. By
Topologies, Lemma \ref{topologies-lemma-fpqc-affine-axioms}
the family $\{Z_{j, l} \to Z\}$ is a standard fpqc covering.
Hence we conclude that $\{X_{ij} \to X\}_{i \in I, j\in J_i}$
is an fpqc covering.
\end{proof}
\begin{lemma}
\label{lemma-recognize-fpqc-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
Suppose that $\{f_i : X_i \to X\}_{i \in I}$ is a family of morphisms of
algebraic spaces with target $X$. Let $U \to X$ be a surjective
\'etale morphism from a scheme towards $X$. Then
$\{f_i : X_i \to X\}_{i \in I}$ is an fpqc covering of $X$ if and only
if $\{U \times_X X_i \to U\}_{i \in I}$ is an fpqc covering of $U$.
\end{lemma}
\begin{proof}
If $\{X_i \to X\}_{i \in I}$ is an fpqc covering, then so is
$\{U \times_X X_i \to U\}_{i \in I}$ by Lemma \ref{lemma-fpqc}.
Assume that $\{U \times_X X_i \to U\}_{i \in I}$ is an fpqc covering.
Let $h : Z \to X$ be a morphism from an affine scheme towards $X$.
Then we see that $U \times_X Z \to Z$ is a surjective \'etale morphism
of schemes, in particular open. Hence we can find finitely many affine opens
$W_1, \ldots, W_t$ of $U \times_X Z$ whose images cover $Z$.
For each $j$ we may apply the condition that
$\{U \times_X X_i \to U\}_{i \in I}$ is an fpqc covering
to the morphism $W_j \to U$, and obtain a standard fpqc covering
$\{W_{jl} \to W_j\}$ which refines $\{W_j \times_X X_i \to W_j\}_{i \in I}$.
Hence $\{W_{jl} \to Z\}$ is a standard fpqc covering of $Z$
(see
Topologies, Lemma \ref{topologies-lemma-fpqc-affine-axioms})
which refines $\{Z \times_X X_i \to X\}$ and we win.
\end{proof}
\begin{lemma}
\label{lemma-refine-fpqc-schemes}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
Suppose that $\mathcal{U} = \{f_i : X_i \to X\}_{i \in I}$ is an
fpqc covering of $X$. Then there exists a refinement
$\mathcal{V} = \{g_i : T_i \to X\}$ of $\mathcal{U}$ which is an
fpqc covering such that each $T_i$ is a scheme.
\end{lemma}
\begin{proof}
Omitted. Hint: For each $i$ choose a scheme $T_i$ and a surjective \'etale
morphism $T_i \to X_i$. Then check that $\{T_i \to X\}$ is an fpqc covering.
\end{proof}
\noindent
To be continued...
\section{Fppf topology}
\label{section-fppf}
\noindent
In this section we discuss the notion of an fppf covering of algebraic spaces,
and we define the big fppf site of an algebraic space. Please compare with
Topologies, Section \ref{topologies-section-fppf}.
\begin{definition}
\label{definition-fppf-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
An {\it fppf covering of $X$} is a family of morphisms
$\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$
such that each $f_i$ is flat and locally of finite presentation
and such that
$$
|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),
$$
i.e., the morphisms are jointly surjective.
\end{definition}
\noindent
This is exactly the same as
Topologies, Definition \ref{topologies-definition-fppf-covering}.
In particular, if $X$ and all the $X_i$ are schemes, then we recover the usual
notion of an fppf covering of schemes.
\begin{lemma}
\label{lemma-fppf}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is an fppf covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is an fppf covering and for each
$i$ we have an fppf covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is an fppf covering.
\item If $\{X_i \to X\}_{i\in I}$ is an fppf covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is an fppf covering.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\begin{lemma}
\label{lemma-refine-fppf-schemes}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
Suppose that $\mathcal{U} = \{f_i : X_i \to X\}_{i \in I}$ is an
fppf covering of $X$. Then there exists a refinement
$\mathcal{V} = \{g_i : T_i \to X\}$ of $\mathcal{U}$ which is an
fppf covering such that each $T_i$ is a scheme.
\end{lemma}
\begin{proof}
Omitted. Hint: For each $i$ choose a scheme $T_i$ and a surjective \'etale
morphism $T_i \to X_i$. Then check that $\{T_i \to X\}$ is an fppf covering.
\end{proof}
\begin{lemma}
\label{lemma-fppf-covering-surjective}
Let $S$ be a scheme.
Let $\{f_i : X_i \to X\}_{i \in I}$ be an fppf covering of algebraic
spaces over $S$. Then the map of sheaves
$$
\coprod X_i \longrightarrow X
$$
is surjective.
\end{lemma}
\begin{proof}
This follows from
Spaces, Lemma \ref{spaces-lemma-surjective-flat-locally-finite-presentation}.
See also
Spaces, Remark \ref{spaces-remark-warning}
in case you are confused about the meaning of this lemma.
\end{proof}
\noindent
To be continued...
\section{Syntomic topology}
\label{section-syntomic}
\noindent
In this section we discuss the notion of a syntomic covering of
algebraic spaces, and we define the big syntomic site of an
algebraic space. Please compare with
Topologies, Section \ref{topologies-section-syntomic}.
\begin{definition}
\label{definition-syntomic-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
A {\it syntomic covering of $X$} is a family of morphisms
$\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$
such that each $f_i$ is syntomic
and such that
$$
|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),
$$
i.e., the morphisms are jointly surjective.
\end{definition}
\noindent
This is exactly the same as
Topologies, Definition \ref{topologies-definition-syntomic-covering}.
In particular, if $X$ and all the $X_i$ are schemes, then we recover the
usual notion of a syntomic covering of schemes.
\begin{lemma}
\label{lemma-syntomic}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is a syntomic covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is a syntomic covering and for each
$i$ we have a syntomic covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a syntomic covering.
\item If $\{X_i \to X\}_{i\in I}$ is a syntomic covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is a syntomic covering.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
To be continued...
\section{Smooth topology}
\label{section-smooth}
\noindent
In this section we discuss the notion of a smooth covering of
algebraic spaces, and we define the big smooth site of an
algebraic space. Please compare with
Topologies, Section \ref{topologies-section-smooth}.
\begin{definition}
\label{definition-smooth-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
A {\it smooth covering of $X$} is a family of morphisms
$\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$
such that each $f_i$ is smooth
and such that
$$
|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),
$$
i.e., the morphisms are jointly surjective.
\end{definition}
\noindent
This is exactly the same as
Topologies, Definition \ref{topologies-definition-smooth-covering}.
In particular, if $X$ and all the $X_i$ are schemes, then we recover the
usual notion of a smooth covering of schemes.
\begin{lemma}
\label{lemma-smooth}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is a smooth covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is a smooth covering and for each
$i$ we have a smooth covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a smooth covering.
\item If $\{X_i \to X\}_{i\in I}$ is a smooth covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is a smooth covering.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
To be continued...
\section{\'Etale topology}
\label{section-etale}
\noindent
In this section we discuss the notion of a \'etale covering of
algebraic spaces, and we define the big \'etale site of an
algebraic space. Please compare with
Topologies, Section \ref{topologies-section-etale}.
\begin{definition}
\label{definition-etale-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
An {\it \'etale covering of $X$} is a family of morphisms
$\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$
such that each $f_i$ is \'etale
and such that
$$
|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),
$$
i.e., the morphisms are jointly surjective.
\end{definition}
\noindent
This is exactly the same as
Topologies, Definition \ref{topologies-definition-etale-covering}.
In particular, if $X$ and all the $X_i$ are schemes, then we recover the
usual notion of a \'etale covering of schemes.
\begin{lemma}
\label{lemma-etale}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is a \'etale covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is a \'etale covering and for each
$i$ we have a \'etale covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a \'etale covering.
\item If $\{X_i \to X\}_{i\in I}$ is a \'etale covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is a \'etale covering.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
The following lemma tells us that the sites
$(\textit{Spaces}/X)_\etale$ and $(\textit{Spaces}/X)_{smooth}$
have the same categories of sheaves.
\begin{lemma}
\label{lemma-etale-dominates-smooth}
Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
Let $\{X_i \to X\}_{i \in I}$ be a smooth covering of $X$.
Then there exists an \'etale covering $\{U_j \to X\}_{j \in J}$
of $X$ which refines $\{X_i \to X\}_{i \in I}$.
\end{lemma}
\begin{proof}
First choose a scheme $U$ and a surjective \'etale morphism $U \to X$.
For each $i$ choose a scheme $W_i$ and a surjective \'etale morphism
$W_i \to X_i$. Then $\{W_i \to X\}_{i \in I}$ is a smooth covering
which refines $\{X_i \to X\}_{i \in I}$. Hence
$\{W_i \times_X U \to U\}_{i \in I}$ is a smooth covering of schemes.
By More on Morphisms, Lemma \ref{more-morphisms-lemma-etale-dominates-smooth}
we can choose an \'etale covering $\{U_j \to U\}$ which refines
$\{W_i \times_X U \to U\}$. Then $\{U_j \to X\}_{j \in J}$
is an \'etale covering refining $\{X_i \to X\}_{i \in I}$.
\end{proof}
\noindent
To be continued...
\section{Zariski topology}
\label{section-zariski}
\noindent
In
Spaces, Section \ref{spaces-section-Zariski}
we introduced the notion of a Zariski covering of an algebraic space by
open subspaces. Here is the corresponding notion with open subspaces
replaced by open immersions.
\begin{definition}
\label{definition-zariski-covering}
Let $S$ be a scheme, and let $X$ be an algebraic space over $S$.
A {\it Zariski covering of $X$} is a family of morphisms
$\{f_i : X_i \to X\}_{i \in I}$ of algebraic spaces over $S$
such that each $f_i$ is an open immersion
and such that
$$
|X| = \bigcup\nolimits_{i \in I} |f_i|(|X_i|),
$$
i.e., the morphisms are jointly surjective.
\end{definition}
\noindent
Although Zariski coverings are occasionally useful the corresponding topology
on the category of algebraic spaces is really too coarse, and not particularly
useful. Still, it does define a site.
\begin{lemma}
\label{lemma-zariski}
Let $S$ be a scheme.
Let $X$ be an algebraic space over $S$.
\begin{enumerate}
\item If $X' \to X$ is an isomorphism then $\{X' \to X\}$
is a Zariski covering of $X$.
\item If $\{X_i \to X\}_{i\in I}$ is a Zariski covering and for each
$i$ we have a Zariski covering $\{X_{ij} \to X_i\}_{j\in J_i}$, then
$\{X_{ij} \to X\}_{i \in I, j\in J_i}$ is a Zariski covering.
\item If $\{X_i \to X\}_{i\in I}$ is a Zariski covering
and $X' \to X$ is a morphism of algebraic spaces then
$\{X' \times_X X_i \to X'\}_{i\in I}$ is a Zariski covering.
\end{enumerate}
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\input{chapters}
\bibliography{my}
\bibliographystyle{amsalpha}
\end{document}